Making sense of non-Hermitian Hamiltonians

          @misc{ pirsa_PIRSA:06040016,
            doi = {10.48660/06040016},
            url = {https://pirsa.org/06040016},
            author = {Bender, Carl},
            keywords = {},
            language = {en},
            title = {Making sense of non-Hermitian Hamiltonians},
            publisher = {Perimeter Institute},
            year = {2006},
            month = {apr},
            note = {PIRSA:06040016 see, \url{https://pirsa.org}}
          }
          

Abstract

It is a standard axiom of quantum mechanics that the Hamiltonian H must be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary. In this talk we examine an alternative formulation of quantum mechanics in which the conventional requirement of Hermiticity is replaced by the more general and physical condition of space- time reflection (PT) symmetry. We show that if the PT symmetry of H is unbroken, Then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are $H=p^2+ix^3$ and $H=p^2-x^4$. Amazingly, the energy levels of these Hamiltonians are all real and positive despite the ``wrong'' sign in the $x^4$ potential! We show that such PT-symmetric Hamiltonians specify physically acceptable quantum-mechanical theories in which the norms of states are positive and time evolution is unitary. To do so we demonstrate that a Hamiltonian that has an unbroken PT symmetry also possesses a new physical symmetry that we call C. Using C, we construct an inner product whose associated norm is positive definite. The result is a new class of consistent complex quantum theories. In effect, we have extended and generalized quantum mechanics into the complex domain. We then discuss PT-symmetric quantum field theories. PT-symmetric scalar field-theoretic Hamiltonians corresponding to the above quantum-mechanical Hamiltonains have interaction terms $ig\phi^3$ and $-g\phi^4$. The latter Theory is interesting because (1) it is asymptotically free and (2) the expectation value of $\phi$ is nonzero. (Thus, such a theory might be useful in describing the Higgs sector.) PT symmetry resolves the long-standing problem of ghosts in the Lee model. When the renormalized coupling constant in this model increases past a critical value, the Hamiltonian ceases to be Hermitian and a negative-norm ghost state appears. At this transition the Hamiltonian becomes PT-symmetric, and the ghost is a physical particle. PT-symmetric QED and the PT-symmetric massive Thirring model will also be discussed. Finally, we mention recent papers which suggest that PT-symmetry may provide insight into cosmological problems.